On abelian 2-categories and derived 2-functors

This is an extended version of my earlier articel “Projective and injective objects in symmetric categorical groups. arXiv:1007.0121v1.” Several new facts added, including the material on the derived 2-functors and the proof of the Gabriel-Mitchel theorem and the Morita theory for 2-rings.

💡 Research Summary

This paper develops a systematic theory of abelian 2‑categories and their derived 2‑functors, extending classical homological algebra into the 2‑dimensional setting. After a brief reminder of Mathieu Dupont’s definition of a 2‑abelian Gp‑d‑category, the author explains that an abelian 2‑category is a groupoid‑enriched 2‑category whose hom‑groupoids are symmetric categorical groups (SCG). Each SCG carries a monoidal addition, and when equipped with a compatible multiplication satisfying Laplaza coherence, it becomes a 2‑ring (also called an Ann‑category or categorical ring).

The paper presents several fundamental examples. The prototypical one is the 2‑category SCG itself. Another is the 2‑category of 2‑modules over a given 2‑ring. A third family of examples is obtained from an ordinary abelian category A by considering arrows A₁ → A₀ as objects. Two variants are defined: A